A body of mass $m$ starts moving from rest along $x$-axis so that its velocity varies as $v = a \sqrt { s }$ where $a$ is a constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in the first $t$ second after the start of the motion is (1) $8 m a ^ { 4 } t ^ { 2 }$ (2) $\frac { 1 } { 4 } m a ^ { 4 } t ^ { 2 }$ (3) $4 m a ^ { 4 } t ^ { 2 }$ (4) $\frac { 1 } { 8 } m a ^ { 4 } t ^ { 2 }$
A thin circular disk is in the $x y$ plane as shown in the figure. The ratio of its moment of inertia about $z$ and $z'$ axes will be: (1) $1 : 4$ (2) $1 : 5$ (3) $1 : 3$ (4) $1 : 2$
An oscillator of mass $M$ is at rest in its equilibrium position in a potential, $V = \frac { 1 } { 2 } k ( x - X ) ^ { 2 }$. A particle of mass $m$ comes from the right with speed $u$ and collides completely inelastic with $M$ and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is: ( $M = 10 , m = 5 , u = 1 , k = 1$ ) (1) $\frac { 2 } { 3 }$ (2) $\frac { 1 } { \sqrt { 3 } }$ (3) $\sqrt { \frac { 3 } { 5 } }$ (4) $\frac { 1 } { 2 }$
Let $p , q$ and $r$ be real numbers ( $p \neq q , r \neq 0$ ), such that the roots of the equation $\frac { 1 } { x + p } + \frac { 1 } { x + q } = \frac { 1 } { r }$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to (1) $p ^ { 2 } + q ^ { 2 }$ (2) $\frac { p ^ { 2 } + q ^ { 2 } } { 2 }$ (3) $2 \left( p ^ { 2 } + q ^ { 2 } \right)$ (4) $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$
If an angle $A$ of a $\triangle A B C$ satisfies $5 \cos A + 3 = 0$, then the roots of the quadratic equation $9 x ^ { 2 } + 27 x + 20 = 0$ are (1) $\sec A , \cot A$ (2) $\sec A , \tan A$ (3) $\tan A , \cos A$ (4) $\sin A , \sec A$
The number of numbers between 2,000 and 5,000 that can be formed with the digits $0,1,2,3,4$ (repetition of digits is not allowed) and are multiple of 3 is (1) 36 (2) 30 (3) 24 (4) 48
The locus of the point of intersection of the lines $\sqrt { 2 } x - y + 4 \sqrt { 2 } k = 0$ and $\sqrt { 2 } k x + k y - 4 \sqrt { 2 } = 0$ ( $k$ is any non-zero real parameter) is (1) an ellipse whose eccentricity is $\frac { 1 } { \sqrt { 3 } }$ (2) a hyperbola whose eccentricity is $\sqrt { 3 }$ (3) a hyperbola with length of its transverse axis $8 \sqrt { 2 }$ (4) an ellipse with length of its major axis $8 \sqrt { 2 }$
If a circle $C$, whose radius is 3 , touches externally the circle $x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 4 = 0$ at the point $( 2,2 )$, then the length of the intercept cut by this circle $C$ on the $x$-axis is equal to (1) $2 \sqrt { 3 }$ (2) $\sqrt { 5 }$ (3) $3 \sqrt { 2 }$ (4) $2 \sqrt { 5 }$
Let $P$ be a point on the parabola $x ^ { 2 } = 4 y$. If the distance of $P$ from the center of the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is (1) $x + y + 1 = 0$ (2) $x + 4 y - 2 = 0$ (3) $x + 2 y = 0$ (4) $x - y + 3 = 0$
If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac { 3 } { 2 }$ units, then its eccentricity is (1) $\frac { 2 } { 3 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 1 } { 9 }$ (4) $\frac { 1 } { 3 }$
If $p \rightarrow ( \sim p \vee \sim q )$ is false, then the truth values of $p$ and $q$ are, respectively (1) $F , F$ (2) $T , T$ (3) $F , T$ (4) $T , F$
The mean and the standard deviation (S. D.) of five observations are 9 and 0 , respectively. If one of the observation is increased such that the mean of the new set of five observations becomes 10 , then their S. D. is (1) 0 (2) 2 (3) 4 (4) 1
A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression of the car to change from $30 ^ { \circ }$ to $45 ^ { \circ }$, then the time taken (in $\min$ ) by the car to reach the foot of the tower is (1) $\frac { 9 } { 2 } ( \sqrt { 3 } + 1 )$ (2) $9 ( \sqrt { 3 } + 1 )$ (3) $18 ( \sqrt { 3 } - 1 )$ (4) $9 ( \sqrt { 3 } - 1 )$